A fixed point formula of Lefschetz type in Arakelov geometry IV: the modular height of C.M. abelian varieties

A fixed point formula of Lefschetz type in Arakelov geometry IV: the modular height of

C.M. abelian varieties

Kai K¨ ohler

Damian Roessler

December 14, 2001

Abstract

We give a new proof of a slightly weaker form of a theorem of P.

Colmez ([C2, Par. 2]). This theorem (Corollary 5.8) gives a formula for the Faltings height of abelian varieties with complex multiplication by a C.M. field whose Galois group overQis abelian; it reduces to the formula of Chowla and Selberg in the case of elliptic curves. We show that the formula can be deduced from the arithmetic fixed point formula proved in [KR2]. Our proof is intrinsic in the sense that it does not rely on the computation of the periods of any particular abelian variety.

1991 Mathematics Subject Classification: 11M06, 14K22, 14G40, 58G10, 58G26

∗Mathematisches Institut, Wegelerstr. 10, D-53115 Bonn, Germany, E-mail:

koehler@math.uni-bonn.de

†Centre de Math´ematiques de Jussieu, Universit´e Paris 7 Denis Diderot, Case Postale 7012, 2, place Jussieu, F-75251 Paris Cedex 05, France, E-mail : roessler@math.jussieu.fr

Contents

1 Introduction 2

2 An arithmetic fixed point formula 4 3 Equivariant geometry on abelian schemes 7 4 The equivariant analytic torsion of line bundles on abelian va-

rieties 10

5 Application of the fixed point formula to abelian varieties with

C.M. by a cyclotomic field 13

1 Introduction

Let A be an abelian variety of dimension d defined over Q. Let K ⊆ C be a number field such that A is defined over K and such that the N´eron model of A over O^K has semi-stable reduction at all the places ofK. Let Ω be the O^K-module of global sections of the sheaf of differentials ofAoverO^K and let α be a section of Ω^d. We write A(C)σ for the manifold of complex points of the varietyA×σ(K)C, whereσ∈Hom(K,C) is an embedding ofKin C. The modular (or Faltings) height ofAis the quantity

hFal(A) := 1

[K:Q]log(#Ω^d/α.Ω^d)− 1 2[K:Q]

X

σ:K→C

log| 1 (2π)^d

Z

A(C)σ

α∧α|. It does not depend on the choice ofKorα. The modular height defines a height on some moduli spaces of abelian varieties and plays a key role in Falting’s proof of the Mordell conjecture. The object of this article is to use higher dimensional Arakelov theory to prove a formula for the modular height ofA, valid ifA has complex multiplication by the ring of integers of an abelian extension of Q. A full self-contained statement of the formula can in be found in corollary 5.8.

This formula was first proved by a completly different method by P. Colmez and the remaining of the introduction is devoted to an exposition of his and our approach, followed by a plan of the paper.

If A is an abelian variety with complex multiplication, the modular height is related to the periods ofA. So suppose that there exists a C.M. fieldE, of degree 2dover Q and an embedding of rings OE →End(A) into the endomorphism ring of A. This is equivalent to saying that A has complex multiplication by OE (cf. [Sh]). We can suppose without loss of generality that the action of OE is defined over K and that K contains all the conjugates of E in C. Let now τ ∈ Hom(E,C) be an embedding of E in C and let ωτ be an element of the first algebraic de Rahm cohomology group H_DR¹ (A) of A over K (this is a K-vector space of rank 2d), such that a(ωτ) = τ(a).ωτ for all a ∈ OE. If σ ∈ Hom(K,C), let ω^σ_τ be the element of the complex cohomology group H¹(A(C)σ,C) obtained by base change. Letuσ be an element of the rational homology groupH1(A(C)σ,Q). The periodP(A, τ, σ)∈C/K^∗ associated toτ and σ∈ Hom(K,C) is the complex number < ω_τ^σ, uσ >∞, where <·,· >∞ is the natural pairing between cohomology and homology. Up to multiplication by an element of K^∗, it is only dependent on τ and σ. Let Φ be the subset of Hom(E,C) such that the subspace {t ∈ T A0 : a(t) = τ(a).t, ∀a ∈ O^E} contains a non-zero element (this is the type of the C.M. abelian variety A).

The following lemma is a (very) weak form of a theorem of P. Colmez:

Lemma 1.1 The equality

(2π)^−d/2.e^{−[K:Q]hFal(A)}= Y

τ∈Φ

Y

σ∈Hom(K,C)

P(A, τ, σ)

holds up to multiplication by an element ofQ.

Furthermore, using a refinement of the above lemma, the theory ofp-adic periods and explicit computation of periods of Jacobians of Fermat curves, he gives an explicit formula forhFal(A) (see also [And], [Yo] and [Gr] [for elliptic curves] for mod. Qversions of the latter formula). To describe it, suppose furthermore that E is Galois over Q and let G:= Gal(E|Q). Identify Φ with its characteristic function G→ {0,1}and define Φ^∨ by the formula Φ^∨(τ) := Φ(τ⁻¹).

Theorem 1.2 (Colmez) If G is abelian, there exists q ∈ Q, such that the identity

1

dhFal(A) =− X

χ odd

<Φ∗Φ^∨, χ >[2L⁰(χprim,0)

L(χprim,0) + log(fχ)] +qlog(2) holds. If the conductor of E over Q divides 8n, where n is an odd natural number, then the identity holds withq= 0.

(it is conjectured that q always vanishes) Here < ·,· > refers to the scalar product of complex valued functions on G and ∗ to the convolution product.

The sum P

χ odd is on all the odd characters of G (recall that χ is odd iff χ(h◦c◦h⁻¹◦τ) =−χ(τ) for allτ, h∈G, wherec∈Gis complex conjugation).

The notationfχ refers to the conductor ofχ. Colmez conjectures that the the- orem 1.2 holds even without the condition thatGis abelian. This formula can be viewed as a generalisation of the formula of Chowla and Selberg (see [CS]), to which it reduces when applied to a C.M. elliptic curve.

It is the aim of this paper to provide a proof of 1.2 using higher dimensional Arakelov theory. More precisely, we shall show that a slightly weaker form of 1.2 can be derived from the fixed point formula in Arakelov theory proved in [KR2] (announced in [KR1]), when applied to abelian varieties with complex multiplication by a field generated over Q by a root of 1. This proof has the advantage of being intrinsic, i.e. the right side of 1.2 is obtained directly from an- alytic invariants (the equivariant analytic torsion and the equivariantR-genus) of the abelian variety. It does not involve the computation of the periods of a particular C.M. abelian variety (e.g. Jacobians of Fermat curves). We shall prove:

Theorem 1.3 Let f be the conductor of E overQ; if Gis abelian there exist numbersap∈Q(µf), wherep|f, such that the identity

1

dhFal(A) =− X

χodd

<Φ∗Φ^∨, χ >2L⁰(χprim,0) L(χprim,0) +X

p|f

aplog(p) holds.

Notice that the difference between the right sides of the equations in 1.2 and 1.3 is equal toP

p|fbplog(p) for somebp∈Q(µf). To see this, let us write X

χodd

<Φ∗Φ^∨, χ >log(fχ) =X

p|f

[X

χodd

<Φ∗Φ^∨, χ > nχ,p] log(p) where nχ,p is the multiplicity of the prime number p in the number fχ. By construction, the number P

χodd < Φ∗Φ^∨, χ > nχ,p is invariant under the action of Gal(Q|Q) and is thus rational.

The paper is organised as follows. In section 2, we give the rephrasing of the main result of [KR2] which we shall need in the text. In section 3, we prove an equivariant version of Zarhin’s trick (this is of independent interest) and we show that the fixed point scheme of the action of a finite group on an abelian scheme is well-behaved away from the fibers lying over primes dividing the or- der of the group. In section 4 we compute a term occuring in the fixed point formula, namely the equivariant holomorphic torsion of line bundles on complex tori; we adapt a method by Berthomieu to do so. In section 5, we apply the fixed point formula to the following setting: an abelian scheme with the action of a certain group of roots of unity and an equivariant ample vector bundle with Euler characteristic 1, which is provided by section 3; an expression for the Faltings height is then very quickly obtained (as a solution of the system of equations (7)) and the rest of the section is concerned with equating this expression with the linear combination of logarithmic derivatives ofL-functions appearing in 1.2.

We shall in this paper freely use the definitions and terminology of section 4 of [KR2] (mainly up to def. 4.1 included).

Acknowledgments. Our thanks go mainly to P. Colmez for many interesting conversations and hints and for kindly providing a proof of corollary 5.3. It is also a pleasure to thank V. Maillot and C. Soul´e for stimulating discussions.

Thanks as well to G. Faltings for pointing out a mistake in an earlier version of this text (a hypothesis was missing in lemma 3.1) and V. Maillot again for pointing out a redundance. We thank the SFB 256, ”Nonlinear Partial Differ- ential Equations”, at the University of Bonn for its support. The second author is grateful to the IHES (Bures-sur-Yvette) for its support in 1998-99.

2 An arithmetic fixed point formula

In this section, we formulate the fixed point formula we shall apply to abelian varieties (more precisely, to their N´eron models). It is an immediate consequence of the more general fixed point formula which is the main result [KR2, Th. 4.4]

of [KR2]. To formulate it, we shall need the notions of arithmetic Chow theory, arithmetic degree and arithmetic characteristic classes; for these see [BoGS,

Par. 2.1]. To understand how it can be deduced from [KR2, Th. 4.4], one needs to read the last section of [KR2]. We shall nevertheless give a self-contained presentation (not proof) of the formula, when the scheme is smooth, and the associated fixed point scheme ´etale over the base scheme.

Let now K be a number field and OK its ring of integers. LetU be an open subset of SpecZ and letUK be the open subset of SpecOK lying above U. Let p1, . . . , pl be the prime numbers in the complement of U. Notice that UK is the spectrum of an arithmetic ring (see the beginning of [KR2, Sec. 4] for the definition). Denote by deg :d dCH¹(UK) → dCH¹(U) the push-forward map in arithmetic Chow theory (this map coincides with the usual arithmetic degree map if UK = SpecO^K). By abuse of language, we shall write deg(Vd ) for the arithmetic degreedeg(dbc1(V)) of the first arithmetic Chern class of a hermitian bundleV onUK. Recall that the formula

ddeg(bc1(V)) =ddeg(bc1(det(V))) = log(#(det(V)/s.UK))− X

σ∈Hom(K,C)

log|s⊗σ(UK)1|^σ

holds. Here s is any section of det(V) and | · |^σ is the norm arising from the hermitian metric of det(V)⊗σ(UK)C. It is a consequence of [GS3, I, Prop. 2.2, p. 13] that there is a canonical isomorphism CHd¹(U)'R/R, where Ris the subgroup ofRconsisting of the expressionsPl

k=1nklogpk, where nk∈Z. We shall thus often identify both.

Let now n∈N^∗ and let C be the subgroup of C generated by the expressions Pl

k=1qkαklogpk, where qk ∈ Q is a rational number and αk is an n-th root of unity. Let f :Y →UK be aµn-equivariant arithmetic variety over UK, of relative dimensiond. As usual, we fix a primitive root of unity ζn. Recall also thatg is the automorphism of the complex manifoldY(C) associated toζn via the action ofµn(C) on Y(C). Fix a g-invariant K¨ahler metric onY(C), with associated K¨ahler formωY.

Theorem 2.1 Let E be a µn-equivariant hermitian vector bundle on Y. Sup- pose thatR^kf∗E= 0fork >0. Then the equality

X

k∈Z/(n)

ζ_n^k.ddeg((R⁰f∗E)k) =

= 1

2Tg(Y(C), E)−1 2 Z

Y_µn(C)

Tdg(T Y)chg(E)Rg(T YC) +degd¡

f_∗^µn(Tdcµn(T f)chbµn(E))¢ holds in C/C.

Recall that R⁰f∗E refers to the UK-module of global sections of E, endowed with the L2-metric inherited from the hermitian metric on E and the K¨ahler

metric onY(C). Recall that the L2-metric is defined as follows: ifs, l are two holomorphic sections ofEC, then

< s, l >L2:= 1 d!(2π)^d

Z

Y(C)

< s, l >Eω_Y^d.

Proof: The proof is similar to the proof of [KR2, Th. 6.14] and so we omit it.

Q.E.D.

The above theorem is an immediate consequence of the arithmetic Riemann- Roch theorem of Bismut-Gillet-Soul´e when the equivariant structure is trivial.

Suppose now thatY is smooth andYµn ´etale overUK. Let L^Im(z, s) :=X

k≥0

Im(z^k) k^s .

where z ∈ C, |z| = 1 and s ∈ C, s > 1. As a function of s, it extends to a meromorphic function of the whole plane, which is holomorphic at s= 0. We write R^rot(arg(z)) for the derivative _∂s^∂ L^Im(z,0) of the functionL^Im(z, s) at 0.

Writeζforζn. We let Ω be the sheaf of differentials off, which is locally free.

Define

D:= Y

k∈Z/n

(1−ζ^k)^{rk (Ωk)}, T = X

k∈Z/n

ζ^krk (Ek)

which are locally constant complex-valued functions on Yµn. In the just men- tionned setting, the formula in theorem 2.1 becomes

X

k∈Z/(n)

ζ_n^k.ddeg((R⁰f∗E)k) =

= 1

2Tg(Y(C), E)−i X

P∈Y_µn(C)

Trace(g|^EP) Det(Id−g|ΩP).[ X

k∈Z/n

rk ((T Y(C)P)k)R^rot(arg(ζ^k))]

+ ddeg¡ f_∗^µn(1

D X

k∈Z/n

ζ^kbc1(Ek) + T D

X

k∈Z/n

ζ^k

1−ζ^kbc1(Ωk))¢

(1) in C/C. Finally, let us recall the following lemma.

Lemma 2.2 Let n ≥1 be a a natural number and let R be an entire ring of characteristic 0 in which the polynomial Xⁿ−1 splits. LetCn be the constant group scheme over Z associated to Z/n, the cyclic group of order n. For each primitiven-th root of unity inR, there is an isomorphism of group schemes

µn×^ZSpecR[1

n]'Cn×^ZSpecR[1 n] overR[¹_n].

Proof: Both group schemes define isomorphic sheaves on the (small) ´etale site of R[_n¹] and they are both ´etale over R[¹_n] (see for instance [Tam, Par. 3.1, p.

100]). Hence they are isomorphic. Q.E.D.

Let as usual Q(µn) refer to the number field generated by the complex n-th roots of unity. In view of the lemma, the constant group scheme Cn and the group scheme µn become isomorphic over Q(µn)[_n¹]. If we letU be the set of prime divisors of n, thenQ(µn)[_n¹] corresponds to the open set UQ(µn). Hence we see that the formula above can be applied to the action of an automorphism of finite order of a (regular, integral, projective) scheme overQ(µn)[_n¹]

3 Equivariant geometry on abelian schemes

The following two lemmata give an equivariant version of Zarhin’s trick. LetK be an algebraically closed field of characteristic zero. IfX is a Z/n-equivariant abelian variety overK andL is a Z/n-equivariant line bundle on X, we shall say that the action ofZ/nonLisnormalised, if it induces the trivial action on the fiberL|0ofLat the origin. In this section, we shall writea(·) for the action of 1∈Z/n.

Lemma 3.1 Let X, Y be Z/n-equivariant abelian varieties defined over K.

Suppose thatZ/nacts by isogenies onX andY. Letf :X →Y be an equivari- ant isogeny and letN be the kernel off. LetM be a line bundle onX. Suppose thatM is endowed with anN-equivariant structure and with a normalisedZ/n- equivariant structure. Let α∈Hom(NZ/n, K^∗)'H¹(NZ/n, K^∗) be defined by the formula α(n) = a⁻¹◦n◦a◦(−n). Then there exists a normalised Z/n- equivariant line bundle L⁰ on Y and aZ/n-equivariant isomorphism f^∗L⁰'L, if and only if α= 1.

Note that we have used the identification Aut(M)'K^∗ in our definition ofα.

Recall also thatNZ/n refers to the part ofN fixed by every element of Z/n.

Proof: If there exists a bundleL⁰satisfying the hypothesies of the lemma, then the equivariant structure off^∗L⁰|NZ/n is by construction trivial and thusα= 1.

So suppose thatα= 1. Note thatN is sent into itself by the elements ofZ/n.

Letρbe the character of N defined by the formula a⁻¹◦a(n)◦a◦(−n) (this character extends α). Since α = 1, ρ induces a character ρ⁰ on the quotient group N/NZ/n. This quotient is naturally identified with the image of the endomorphisma−Id ofN. Viewρ⁰as a character on Im(a−Id) and choose any characterρ⁰⁰extending (ρ⁰)⁻¹toN. Such a character always exists becauseK^∗ is an injective abelian group. We modify the naturalN-equivariant structure on M by multiplying byρ⁰⁰(n) the automorphism ofM given byn, for eachn∈N.

With this new structure, we have the identitya(n)◦a=a◦nof automorphisms

of (the total space of)M. To see this consider the identities (a⁻¹◦(a(n).ρ⁰⁰(a(n)))◦a)◦(−nρ⁰⁰(−n))

= (a⁻¹◦a(n)◦a◦(−n)).ρ⁰⁰(a(n)).ρ⁰⁰(−n)

= (a⁻¹◦a(n)◦a◦(−n)).ρ(n)⁻¹= Id

Consider now the bundle (f∗M)^N, which is the quotient of the bundle M by the action of N. Using the identitya(n)◦a=a◦n, we see that the action of a descends to (f∗M)^N. Furthermore, it is shown in [Mu, Prop. 2, p. 70] that (f∗M)^N is naturally isomorphic to the total space of a line bundle onY =X/N.

By [Mu, Prop. 2, p. 70] again, this bundle has the required properties. Q.E.D.

Lemma 3.2 Let A be a Z/n-equivariant abelian variety over K, where Z/n acts by isogenies. There exists aZ/n-equivariant abelian varietyB overK and an ampleZ/n-equivariant bundleL onB, such that

(a) B is (non-equivariantly) isomorphic to(A×A^∨)⁴;

(b) the equationχ(L) = 1 holds for the Euler characteristic ofL;

(c) the groupZ/nacts by isogenies onB and there exists an equivariant isogeny pfrom A⁸ toB.

Proof: We shall follow the steps of the proof of Zarhin’s trick. LetP⁰ be any ample line bundle on A. Let P :=⊗g∈Z/ng^∗P⁰. The bundle P is ample and carries a naturalZ/n-equivariant structure. Endow A⁴ with the induced Z/n- equivariant structure and let pi :A⁴ →A be the i-th projection (i= 1, . . .4).

Let M⁰ =p^∗₁P ×p^∗₂P×p^∗₃P ×p^∗₄P be the fourth external tensor power of P.

This is again an ampleZ/n-equivariant line bundle onA⁴. Letmbe the order of the Mumford groupK(M⁰) of M⁰ (see [Mu, p. 60] for the definition). Now leta, b, c, dbe integers such thata²+b²+c²+d²=−1(modm²). Consider the endomorphism α(m) ofA⁴ described by the matrix







a −b −c −d

b a d −c

c −d a b

d c −b a







(this is the endomorphism appearing in the proof of Zarhin’s trick). This endo- morphism commutes with all the elements ofZ/n, becauseZ/nacts by isogenies.

Let nowN be the subgroup ofX =A⁸given by the graph ofα(m)|K(M⁰). This subgroup is sent into itself by all the elements ofZ/n. Let B =X/N and let p : X → B be the quotient map. By construction B carries a natural Z/n- equivariant structure such thatpis equivariant. It is shown in [Mi, Rem. 6.12,

p. 136] that there exists a line bundleLonBand an (non-equivariant) isomor- phismM⁰⊗ExtM⁰ 'p^∗L; we can thus endowM⁰⊗ExtM⁰ with anN-equivariant structure. TheZ/n-equivariant structure onM⁰⊗ExtM⁰|XZ/n is by construction trivial. By the last lemma, we can thus assume thatLcarries aZ/n-equivariant structure and that there is anZ/n-equivariant isomorphismM⁰⊗ExtM⁰'p^∗L.

We claim thatBandLare the objects required in the statement of the lemma.

The fact that (a) holds is a step in the proof of Zarhin’s trick and we refer to [Mi, Rem. 6.12, p. 136] for the details. To see that (b) holds, we use [Mu, Th.

2, p. 121] and compute χ(M) =χ(M⁰)² = #N and χ(M) = #N.χ(L), from which we deduce that χ(L) = 1. To see that (c) holds, note that the map p defined above in the proof has the properties required ofpin (c). Q.E.D.

We now quote the following results on extensions of line bundles from the generic fiber. IfX →S is any S scheme,L is a line sheaf overX andi:S→X is an S-valued point, then a rigidificationof Lalongi is an isomorphismi^∗L' OS. The bundle L together with a rigidification will be said to be rigidifiedalong i. Once a point is given, the line bundles rigidified along that point form a category, where the morphisms are the sheaf morphisms that commute with the rigidification.

Proposition 3.3 Let A →SpecK be an abelian variety over a number field.

Suppose thatAhas good reduction at all the finite places ofK. LetA →SpecOK

be its N´eron model. LetRA(resp. RA) be the category of line bundles rigidified along the 0-section ofA(resp. A).

(1) (see [MB, 1.1, p. 40]) The restriction functorRA→RA is an equivalence of categories;

(2) (see [Ra, Th. VIII.2]) ifLis an ample line bundle onA, then any extension ofL toAis ample.

Now letKbe a number field that contains all then-th roots of unity. LetAbe an abelian variety overKthat has good reduction at all the finite places ofK.

LetAbe its N´eron model over OK and letA⁰ :=A ×SpecOKSpecOK[¹_n]. Let f :A⁰ →SpecOK[1/n] be the structure map and let Ω := ΩA⁰ be the sheaf of differentials off.

Suppose thatA⁰is endowed with an action ofZ/nby SpecOK[¹_n]-group scheme automorphisms. Let as usualf^Z/n :A⁰Z/n →SpecOK[_n¹] be the structure map of the fixed scheme. Let i0 : SpecO^K[_n¹] → A⁰ be the zero section. We let 1∈Z/n act bya(·), as before.

Lemma 3.4 If i^∗₀Ωhas noO^K[1/n]-submodule, which is fixed under the action of Z/n, then the schemeA⁰Z/n is ´etale and finite overSpecO^K[¹_n].

Proof: We first prove that f^Z/n is ´etale. We have to show that f^Z/n is flat and that for any geometric pointx→SpecOK[1/n], the corresponding scheme obtained by base-change is regular. The second condition is verified because of [KR2, Cor. 2.9] (base-change invariance of the fixed scheme) and [KR2, Prop. 2.10] (regularity of the fixed-scheme). To show that f^Z/n is flat, we apply the criterion [Ha, Th. 9.9, p. 261]. Choose a very ampleZ/n-equivariant line bundle LonA⁰. Let pbe a maximal ideal of SpecOK[1/n] and denote by k(p) the corresponding residue field. The Hilbert-Samuel polynomial of the fiber A⁰Z/n,k(p)ofA⁰Z/natprelatively toL|A⁰_Z/n,k(p) can be computed on the algebraic closurek(p) ofk(p). Asi^∗₀Ω has no fixed part, we see thatA⁰Z/n(k(p)) consists of isolated fixed points only. The numberF(p) of these fixed points is the Hilbert- Samuel polynomial of A⁰Z/n,k(p), which is of degree 0. We have to show that F(p) is independent ofp. To prove this, consider first thatH¹(A⁰_k(p),Lk(p)) = 0 for allp; this follows from the characterization of the cohomology of line bundles on abelian varieties [Mu, Par. 16, p. 150]. Thus we know thatf∗Lis locally free and that there are natural equivariant isomorphisms (f∗L)k(p) 'fk(p),∗Lk(p). We may also assume that the action onL|A⁰_Z/n is trivial; this might be achieved by replacing L by its n-th tensor power. Using the Lefschetz trace formula of [BFQ], we see thatF(p) depends only on the trace ofaonH⁰(AK,LK) and on the determinant of Id−aoni^∗_0,KΩK (where (·)K refers to the base change by Spec K →SpecOK[_n¹]). ThusF(p) is independent of p. To see that f^Z/n is finite, we only have to check that it is quasi-finite, as it is projective (see [Ha, Ex.

11.2]). Let againpbe a prime ideal. Asf^Z/n is ´etale, we know thatA⁰Z/n,k(p)is the spectrum of a direct sum of finite field extensions ofk(p); furthermore this sum is finite, since the morphism is of finite type. Hence A⁰Z/n,k(p) is a finite set and thus we are done. Q.E.D.

4 The equivariant analytic torsion of line bun- dles on abelian varieties

Let (V, g^V) be a d-dimensional Hermitian vector space and let Λ ⊂ V be a lattice of rank 2d. The quotient A:=V /Λ is a flat complex torus. According to the Appell-Humbert theorem [LB, Ch. 2.2], the holomorphic line bundles on A can be described as follows: Choose an Hermitian form H on V such that E:= ImH takes integer values on Λ×Λ. Choose furthermoreα: Λ→S¹such that

α(λ1+λ2) =α(λ1)α(λ2)e^{πiE(λ1,λ2)}

for allλ1, λ2 ∈Λ. Then there is an associated line bundle LH,α defined as the quotient of the trivial line bundle onV by the action of Λ given by

λ◦(v, t) := (v+λ, α(λ)eπH(v,λ)+πH(λ,λ)/2t).

There is a canonical Hermitian metrich^L onLH,α given by h^L((v, t1),(v, t2)) :=t1¯t2e^−πH(v,v) .

Define C ∈ End(V) by H(v1, v2) =g^V(v1, Cv2). C is Hermitian with respect to g^V. Consider an automorphismg of (V, g^V), leaving Λ, H and αinvariant.

ThengandCcommute, thus they may be diagonalized simultaneously. Denote their eigenvalues by (e^iφj) and (νj), respectively, and let (ej) be a corresponding set ofg^V-orthonormal eigenvectors. Assume that for allj, φj ∈/ 2πZ, i.e. that g acts on A with isolated fixed points. Note that the isometric automorphism g ofAhas finite order.

LetLTr(LH,α) denote the trace of the action ofgonH⁰(A, LH,α).

Lemma 4.1 Assume that νj = 0 for all j. Then the equivariant analytic tor- sion of L¯H,α on (A, g^V) with respect tog vanishes.

Proof: LetV_R^∨ be the dual of the underlying real vector space ofV. Consider the dual lattice Λ^∨ :={µ∈V_R^∨|µ(λ)∈2πZ∀λ∈Λ}. Representα=e^iµ0 with µ0 ∈V_R^∨. It is shown in [K4, section V] that the eigenfunctions are given by the functions fµ : VR/Λ → C, x7→e^i(µ+µ0)(x) with corresponding eigenvalue

1

2kµ+µ0k². The eigenforms are given by the product of these eigenfunctions times the pullback of elements of Λ^∗V^∨. The eigenvalue of such a formfµ·η is the eigenvalue offµ.

As LH,α is g-invariant, we get g^∗e^iµ0 = e^i(µ0+µ1) for some µ1 ∈ Λ^∨. Also, g maps Λ^∨ to itself, thus for anyµ∈Λ^∨ there is someµ⁰ ∈Λ^∨with g^∗fµ=fµ⁰. As g acts fixed point free on V \ {0}, it maps a function fµ to a multiple of itself iffµ+µ0= 0, i.e. ifffµ represents an element in the cohomology. Thusg acts diagonal free on the complement of the cohomology, and the zeta function defining the torsion vanishes. Q.E.D.

Letψ:R/Z→Rdenote the functionψ([x]) := (log Γ)⁰(x) forx∈]0,1].

Theorem 4.2 Assume that νj > 0 for all j. Then the equivariant analytic torsion of L¯H,α on (A, g^V)with respect tog is given by

Tg(A,L¯H,α) = Xd j=1

"

log(2πνj)

e^−iφj −1 +iR^rot(φj)

−1 4

µ

2 log(2π)−2Γ⁰(1) +ψ([φj

2π]) +ψ([1−φj

2π])

¶ #

LTr(LH,α).

Proof: Assume first that theνj are pairwise linear independent overQ. Then as is shown in [Ber, p. 3], the spectrum of the Kodaira-Laplace operator onA

is given by

σ(¤) =© 2π

Xd j=1

njνj

¯¯n_j ∈N0 ∀jª .

Seteⁱ :=g^V(ei,·). LetE^q_νdenote the eigenspace corresponding to the eigenvalue ν of¤on Γ^∞(A,Λ^qT^∗A⊗LH,α). We shall prove by induction that the trace of g onE^q_2π^P

jnj|νj|is given by µ#{j|nj 6= 0}

q

¶

LTr(LH,α)·Y

j

e^injφj . (2)

First this formula holds forE⁰₀. Now the eigenspaces verify the relations E^q_ν= M

j1<...<jq

³Λ^q_k=1e^jk⊗E_ν−2π^{0 Pq}

k=1ν_jk

´ (3)

[Ber, eq. (7)] and the complex

0→E_ν⁰→ · · ·^∂¯ →^∂¯ E_ν^d→0 (4) is acyclic. Thus we can determine the trace on E_ν^q forq > 0 by the trace on some E_ν⁰0 withν⁰ < ν. Then the trace onE_ν⁰ is given by the trace on theE_ν^q forq >1 by the sequence 4. As the relations (3), (4) are compatible with (2), equation (2) is proven.

Hence the zeta function defining the torsion is given by Z(s) =

Xd q=0

X

ν∈σ(¤)

(−1)^q+1q ν^s

µ#{j|nj 6= 0} q

¶

LTr(LH,α)·Y

j

e^injφj .

Notice that for any 0≤k≤n Xd q=0

(−1)^q+1q µk

q

¶

=

½ 1 fork= 1 0 fork6= 1 .

Consider forz∈C,|z|= 1 and fors∈C, Res >1 the zeta function L(z, s) :=

X∞ k=1

z^k k^s

and its meromorphic continuation tos∈C. Thus Z(s) =

Xd j=1

X∞ k=1

e^ikφj

(2πkνj)^sLTr(LH,α) = Xd j=1

(2πνj)^−sL(e^iφj, s)LTr(LH,α)

and

Z⁰(0) = Xd j=1

¡−log(2πνj)L(e^iφj,0) +L⁰(e^iφj,0)¢

LTr(LH,α).

Using the definition ofR^rot(φ) := _∂s|s=0^∂ _2i¹(L(e^iφ, s)−L(e^−iφ, s)) in [K1] (com- pare section 2), the formula for _∂s^∂ _|s=0¹₂(L(e^iφ, s) +L(e^−iφ, s)) in [K3, Lemma 13] and the formula L(e^iφ,0) = (e^−iφ−1)⁻¹ (e.g. in [K2, p. 108]) we find the theorem forνj pairwise linear independent overQ. If the cohomology does not change, the torsion varies continuously with the metric ([BGS3, section (d)]), thus the result holds for any nonzeroνj. Alternatively, one can show that for- mulas (3), (4) hold more general by arguing as in [Ber, Remark p. 4]) or by continuity. Q.E.D.

Remark. More general, assume only that allνj are non-zero. Again, by the results of [Ber] and a proof similar to the above one we get

Tg(A,L¯H,α) = Xd j=1

sign(νj)

"

log(2π|νj|)

e^−iφj −1 +iR^rot(φj)

−1 4

µ

2 log(2π)−2Γ⁰(1) +ψ([φj

2π]) +ψ([1−φj

2π])

¶ #

LTr(LH,α). Combining this with Lemma 4.1 as in [Ber, section 4] by splittingA(see also [LB, chapter 3,§3]) and using the product formula for equivariant torsion [K2, Lemma 2], one obtains the value of the equivariant torsion for any LH,α. Similarly, for automorphismsghaving a larger fixed point set one can split Aaccordingly to obtain the value of the torsion.

5 Application of the fixed point formula to abelian varieties with C.M. by a cyclotomic field

Let now n > 0 be a natural number and let φ(n) := #(Z/(n))^×. Let A be an abelian variety of dimension d = φ(n)/2 defined over a number field K.

Suppose that A_Q has complex multiplication by OQ(µn) and fix a ring em- bedding OQ(µn) → End(A_Q) (see [Sh] for the general theory). As before we choose a primitiven-th root of unity ζ:=ζn; this root defines an isomorphism µn(C) 'Z/n and thus a canonicalZ/n-action on A_Q. We may suppose that Q(µn)⊆K and thatA has good reduction at all the finite places of K. The latter hypothesis is possible in view of [ST, Th. 7, p. 505]. Let B be the abelian variety obtained fromA_Q via lemma 3.2. We can suppose without loss of generality that B, as well as the line bundle L promised in 3.2, are defined overK; we may also suppose that the mappappearing in (c) is defined overK.

The existence ofpshows thatB has good reduction at the finite places ofK as well. Furthermore, we normalize the action ofZ/nonLso that its restriction to L|0 becomes trivial (one might achieve this by multiplying the action by some character of Z/n). Choose some rigidification of L along the 0-section. The action ofZ/n will then leave this rigidification invariant.

We let π : B → SpecOK be the N´eron model of B. By the universal prop- erty of N´eron models, the action of Z/n on B extends to B. Also, by (1) of Proposition 3.3, there exists a line bundle LonB, endowed with a Z/n-action and a Z/n-invariant rigidification along the 0-section, which both extend the corresponding structures onL. By [MB, Prop. 2.1, p. 48], there exists a unique metric on L^C whose first Chern form is translation invariant and such that the rigidification is an isometry. We endow LC with this metric. It is Z/n- invariant by unicity. We endow B(C) with the translation invariant K¨ahler metric given by the Riemann form of LC. We let Ωπ be the sheaf of differ- entials of π and endow it with the metric induced by the K¨ahler metric. If we let i0 : SpecO^K → B be the 0-section, we then have a canonical isometric equivariant isomorphismπ^∗i^∗₀Ωπ 'Ωπ. It is shown in [Bo, Par. 4, 4.1.10] that deg(id ^∗₀Ωπ) = [K:Q]hFal(B_Q).

We now apply the equivariant arithmetic Riemann-Roch formula (1) toL. We shall work overOK[1/n], over whichZ/n- andµn-action are equivalent concepts because of lemma 2.2. LetdB= 8dbe the relative dimension ofB. Letp1, . . . , pl

be the prime factors ofn and let U be the complement of the set {p1, . . . , pl} in SpecZ. We then have the identification UK ' SpecO^K[1/n]. We identify µn and Z/n over UK via the root of unity ζ. We letS be the subgroup of R generated by the expressionsPl

k=1qkIm(αk) logpk, whereqk ∈Qis a rational number and αk is ann-th root of unity.

Proposition 5.1 Let e^iφ1, . . . e^iφdB be the eigenvalues of the action of1∈Z/n on T B|0 (with multiplicities). The equality

1 [K:Q]

X

k∈Z/n

Im( ζ^k

1−ζ^k)ddeg(i^∗₀Ωπ,k) =1 2

dB

X

j=1

R^rot(φj) (5) holds in R/S.

Proof: For any l∈Z, let [l] denote thel-plication map from a group scheme into itself. First notice that by [Bo, (4.1.23)] (i.e. a hermitian refinement of the theorem of the square), the identity [l]^∗bc1(L ⊗[−1]^∗L) =l²bc1(L ⊗[−1]^∗L) holds in dCH¹(B) , for everyl∈Z. Now consider that by lemma 3.4, the schemeBµn

is a finite commutative group scheme overOK[1/n]; it thus has an orderl0≥0 and the l0-plication map is the 0-map on Bµn (see [Ta]). Now we compute in dCH¹(B^µn):

[l0]^∗bc1(L|^Bµn⊗[−1]^∗L|^Bµn) =l²₀bc1(L|^Bµn⊗[−1]^∗L|^Bµn) = 0

and thus we getdegd¡

f∗^µn(l₀²bc1(L|^Bµn)+l²₀bc1([−1]^∗L|^Bµn))¢

=degd¡

f∗^µn(2.l²₀bc1(L|^Bµn))¢

= 0 and finallyddeg¡

f∗^µn(bc1(L^Bµn))¢

= 0.

Let us now write down the equation (1) in our situation. First notice that in view of the equivariant isomorphismπ^∗i^∗₀Ωπ 'Ωπ, the functions T andD are constant. Until the end of the proof, let N be the number of fixed points of the action of 1 on an arbitrary connected component of B(C). The holomor- phic Lefschetz trace formula (see [BFQ] or [ASe, III, (4.6), p. 566]) shows that N.T /D=P

k∈Z/nζ_n^krk ((R⁰f∗L)k) and thusN is independent of the choice of the component. We obtain

X

k∈Z/(n)

ζ^k.ddeg((R⁰f∗L)k) =

= 1

2Tg(B(C), E)−i[K:Q]N.T D .

dB

X

k=1

R^rot(φk) + degd¡

f_∗^µn(T D. X

k∈Z/n

ζ^k

1−ζ^kbc1(Ωπ,k))¢

in C/C. As rk (R⁰π∗(L)) = 1, we also have the equality of complex numbers X

k∈Z/(n)

ζ^k.ddeg((R⁰f∗L)k) =ddeg((R⁰f∗L)).N.T D

And thus we get

ddeg((R⁰f∗L))

= (N.T D )⁻¹1

2Tg(B(C), E)−i[K:Q]

dB

X

k=1

R^rot(φk) + degd¡

f_∗^µn( X

k∈Z/n

ζ^k

(1−ζ^k)bc1(Ωπ,k))¢

.N⁻¹ (6)

inC/C. The theorem 4.2 shows that the imaginary part of (^N.T_D )^{−1 1}₂Tg(B(C), E) is equal to [K:Q]R^rot(φk). Furthermore, using the isomorphismπ^∗i^∗₀Ωπ'Ωπ

and the projection formula, we see that ddeg¡

f_∗^µn( X

k∈Z/n

ζ^k

1−ζ^kbc1(Ωπ,k))¢

.N⁻¹= X

k∈Z/n

ζ^k

1−ζ^kbc1(i^∗₀Ωπ,k).

Taking these two facts into account, we can take the imaginary part of both sides in (6) to conclude. Q.E.D.

Let now Φ := {Φ1, . . . ,Φd} be the type of A_Q. Define ζ^P(k) = Φk(ζ) for k= 1, . . . , d. In view of 3.2 (c), the identity (5) can be rewritten as

1 [K:Q]

Xd k=1

Im( Φk(ζ)

1−Φk(ζ))ddeg(i^∗₀Ωπ,P(k)) =−4 Xd k=1

∂

∂sL^Im(Φk(ζ),0) (inR/S). Now notice that we can change our choice ofζin the latter equation and replace it byσ(ζ), whereσ∈Gal(Q(µn)|Q) (this corresponds to applying the fixed point formula to a power of the original automorphism), thus obtain- ing a system of linear equations in the deg(id ^∗₀Ωπ,k). Notice also the equations obtained fromσ(ζ) andσ(ζ) are equivalent. With this remark in mind, we see that the just mentioned system of equations is equivalent to the following one:

1 [K:Q]

Xd k=1

Im( Φ⁻¹_k ◦Φl(ζ)

1−Φ⁻¹_k ◦Φl(ζ))Xk =−4 Xd k=1

∂

∂sL^Im(Φ⁻¹_k ◦Φl(ζ),0) +El (7) (inR) wherel= 1, . . . , dand the coefficients of the vectorE:=^t[E1, . . . , Ed] lie in S. We shall show that the matrix M := [Im( ^{Φ−1k ◦Φl(ζl)}

1−Φ⁻¹_k ◦Φl(ζ))]l,k of this system is invertible as a matrix of real numbers. Since the coefficients of M all lie in Q(µn), the coefficients of M⁻¹ lie in Q(µn) as well and thus we see that the coefficients of the vectorM⁻¹E lie in C. Thus we can determine the quantities Xk up to an element ofS. Now recall that by construction

hFal(BQ) = 1 [K:Q](

Xd k=1

deg(id ^∗₀Ωπ,P(k))).

inR/S. Furthermore, by a result of Raynaud in [SCM, Exp. VII, Cor. 2.1.3, p.

207], the modular height of an abelian variety and the modular height of the dual of the latter are equal. Thus, using (a) 3.2, we see that hFal(BQ) = 8.hFal(A_Q) (inR). Thus we can determine hFal(A_Q) up to an element ofS.

Before we proceed to solve the system (7), we need two lemmata that relate the quantities appearing in the system with Dirichlet L-functions.

Warning. In what follows, in contradiction with classical usage, the notation L(χ, s) will always refer to the non-primitive L-function associated with a Dirichlet character. We writeχprim for the primitive character associated with χand accordingly write L(χprim, s) for the associated primitive L-function.

First take notice of the elementary fact that Im(z/(1−z)) = ¹₂cot(¹₂arg(z)) if |z| = 1. Let G be the Galois group of the extension Q(µm)|Q. From now on, for simplicity we fix ζ = e^2πi/n. The following lemma is a variation on the functional equation of the Dirichlet L-functions. When χ is a primitive character, it can be derived directly from the functional equation and classical results on Gauss sums.

Lemma 5.2 Let χ be an odd character ofG. The equality

< L^Im(σ(ζ), s), χ >:= 1 2d

X

σ∈G

χ(σ)L^Im(σ(ζ), s) = 1

2dn^1−s Γ(1−s/2)

Γ((s+ 1)/2)π^s−1/2L(χ,1−s) holds for all s∈C.

In the expression< L^Im(σ(ζ), s), χ >, the symbolσ is considered as a variable in G.

Proof: We prove the equality for 0< s <1. The full equality then follows by analytic continuation. We compute

n^s+12 Γ(s+ 1

2 )π^−s+12 1 2d

X

σ∈G

χ(σ)L^Im(σ(ζ), s)

= n^s+12 Γ(s+ 1

2 )π^−s+12 1 2d

X

σ∈G

X

k≥0

Im(σ(ζ)^k) k^s χ(σ)

= n^s+12 Γ(s+ 1

2 )π^−s+12 1 2d

X

k≥0

1 k^s

X

σ∈G

χ(σ)Im(σ(ζ)^k)

= X

k≥0

Z ∞ 0

1

2dk.e^−k2πu/nu^12(s+1)−1[X

σ∈G

χ(σ)Im(σ(ζ)^k)]du

= −i.

Z ∞ 0

1 2d

X

k≥0

k.e^−k

2π

nu u^{−1−12(s+1)}[X

σ∈G

χ(σ)σ(ζ)^k]du (8)

= Z ∞

0

1 2d

X

k≥0

χ(k)u^{−1−12(s+1)}k

ne^−πk2u/n(n.u)³²du (9)

= 1

2d X

k≥0

χ(k) Z ∞

0

u^{−1−12(s+1)}k

ne^−πk2u/n(n.u)³²du (10)

= 1

2dn^3/2−s/2Γ(1−s/2)π^s/2−1L(χ,1−s)

For the equality (9), we used the Poisson summation formula. To obtain the equality (8), we first exchange the summation and integration symbols and then make the change of variable u 7→ _u¹. The exchange of summation and integration symbols is justified by the following estimates. Letσ0 ∈G and let tk := Im(σ0(ζ)^k) and vk := k.e^k2πu/n. Since Pln

k=0tk = 0 for all l ≥ 0, the sequenceTk:=Pk

j=0tj is bounded above byC >0. Consider now that

| XN k=0

k.e^−k2πu/nIm(σ0(ζ)^k)|=| XN k=0

tkvk|=|

NX−1 k=0

Tk(vk−vk+1) +TNvN| (11) where we used partial summation for the last equality. The function (of k) k.e^−k2πu/nis increasing on the interval [0,p n

2πu] and decreasing on the interval

[p n

2πu,∞[. Letk0be the largest integer less or equal top n

2πu. The expression (11) can be bounded above by

CvN +C

N−1X

k=k0+1

(vk−vk+1) +C|vk0−vk0+1|+C

kX0−1 k=0

(vk+1−vk)

= CvN+C(vk0+1−vN) +C(vk0−v0) +C|vk0−vk0+1|=C[vk0+1+vk0+|vk0−vk0+1|]

= 2Cmax{vk0+1, vk0} ≤2C r n

2πeu^−1/2. Hence

| XN k=0

k.e^−k2πu/nu^12(s+1)−1Im(σ0(ζ)^k)| ≤2C r n

2πe|u^−1/2u^12(s+1)−1|= 2C r n

2πe|u^12s−1| On the other hand, foru >1, the classical estimate

| XN k=0

k.e^−k2πu/nu^12(s+1)−1Im(σ0(ζ)^k)| ≤ | XN k=0

k.e^−kπu/nu^12(s+1)−1)| ≤

≤e^−πu/n|u^12(s+1)−1| 1 (1−e^−πu/n)²

holds. The first estimate show that for u∈]0,1[ ands >0, the function ofu

| XN k=0

k.e^−k2πu/nu^12(s+1)−1Im(σ0(ζ)^k)|

is bounded by an element of L¹([0,1]). The second estimate shows that for u∈]1,∞[ ands∈Cthe same function is bounded by an element ofL¹(]1,∞[).

By the dominated convergence theorem, we might thus exchange summation and integration symbols. Completly similar estimates justify the equality (10).

Q.E.D.

It is shown in [K1, Proof of Th. 8, p. 564] thatL^Im(σ(ζ),0) = ¹₂cot(¹₂arg(σ(ζ))).

Thus, we see that the last lemma has the following corollary.

Corollary 5.3 For any odd character χon G, the equality X

σ∈G

cot(1

2arg(σ(ζ)))χ(σ) = 2n π L(χ,1) holds.

To tackle with the system (7), we shall need the following result from linear algebra. We say that a complex-valued functionf onGisoddiff(c◦x) =−f(x) for allx∈G, where cdenotes complex conjugation.

Lemma 5.4 Let X~ := (X1, . . . , Xd) and Y~ := (Y1, . . . , Yd). Let f be any odd function onGand let Mf be the matrix [f(Φ⁻¹_k ◦Φl)]l,k. If< χ, f >6= 0 for all the odd charactersχ, then the system M ~X=Y~ is maximal and

Xj = X

χodd

χ(Φj)P

lχ(Φl)Yl

d²< f, χ >

Proof: Letφk be the function defined onGsuch thatφk(x) = 1 ifx= Φ1◦Φk

and 0 otherwise (k= 1, . . . , d). onG. LetV⁻be the complex vector space of odd functions onG. An ordered basisBΦofV⁻is given byφ1−φ1◦c, . . . , φd−φd◦c.

Another basis Bχ is given by the odd characters onG. We viewM as a linear endomorphism of V⁻, viaBΦ. We now proceed to find the matrix ofM in the basisBχ. We compute

M.^t[χ(Φ1◦Φ1), . . . , χ(Φ1◦Φd)] =^t[X

k

f(Φ⁻¹_k ◦Φl)χ(Φ1◦Φk)]l

and

X

k

f(Φ⁻¹_k ◦Φl)χ(Φ1◦Φk) = 1 2

X

σ∈G

f(σ⁻¹◦Φl)χ(Φ1◦σ)

= 1

2 X

σ∈G

f(σ⁻¹)χ(σ◦Φl◦Φ1) =d < f, χ > χ(Φ1◦Φl).

Thus M is represented by the diagonal matrix Diag[d < f, χ >]χ in the basis Bχ. The vectorY~χ :=¹_d[P

lYkχ(Φ1◦Φl)]χrepresents the vectorY~ inBχ. Thus the solution ofM ~X =Y~ inBχis the vectorX~χ:= [

P

lYkχ(Φ1◦Φl)

d²<f,χ> ]χand we thus obtain

Xj= X

χ odd

χ(Φ1◦Φj) P

lYkχ(Φ1◦Φl) d²< f, χ > = X

χodd

χ(Φj)P

lχ(Φl)Yl

d²< f, χ > (12) Q.E.D.

In the following proposition, we apply the lemma 5.4 to the system (7) and use the lemmata 5.3 and 5.2 to evaluate the resulting expression in terms of logarithmic derivatives ofL-functions.

Proposition 5.5 The Faltings height of A_Q is given by the identity 1

dhFal(A) =− X

χodd

<Φ∗Φ^∨, χ >2L⁰(χprim,0) L(χprim,0) +X

p|n

aplog(p) whereap∈Q(µn).

Proof: We apply the lemma 5.4 to the system (7) (with El = 0). Define f :G→C by the formulaf(σ) := cot(¹₂arg(σ(ζ))). The fact that the system is maximal is implied by the fact that L(χ,1) 6= 0, when χ is a non-principal Dirichlet character (see for instance [CaFr, Th. 2, p. 212]). We compute

1 8

X

j

Xj=−X

j

X

χ odd

χ(Φj)P

k,l ∂

∂sL^Im(Φ⁻¹_k ◦Φl(ζ),0)χ(Φl) d²< f, χ >

= −X

χ

P

jχ(Φj)P

k,l ∂

∂sL^Im(Φ⁻¹_k ◦Φl(ζ),0)χ(Φl) d²< f, χ >

Using scalar and convolution products, we can write P

jχ(Φj)P

k,l

∂

∂sL^Im(Φ⁻¹_k ◦Φl(ζ),0)χ(Φl) d²< f, χ >

= 2d < χ,Φ> .2d²< χ,_∂s^∂ L^Im(σ(ζ),0)∗Φ∗(·)⁻¹>

d²< f, χ >

= 4d³< χ,Φ>< χ,_∂s^∂L^Im(σ⁻¹(ζ),0)∗Φ^∨>

d²< f, χ >

= 8d²< χ,Φ∗Φ^∨> .< _∂s^∂L^Im(σ(ζ),0), χ >

2n πL(χ,1)

= 4d²π

n <Φ∗Φ^∨, χ > .<_∂s^∂ L^Im(σ(ζ),0), χ >

L(χ,1) (13)

where σ∈Gis a variable and the equalities are inR. Now by lemma 5.2, the sum over all oddχof (13) is equal to

X

χ odd

<Φ∗Φ^∨, χ >( 4d²π n

∂

∂s[L(χ,1−s)n^1−s_Γ((s+1)/2)^Γ(1−s/2) π^s−1/2]s=01 2d

L(χ,1) ) (14)

Furthermore, from the existence of Euler product expansions for L functions, we deduce that

L⁰(χ,1)

L(χ,1) =L⁰(χ_prim,1) L(χ_prim,1) +X

p|n

χ_prim(p)/p

1−χ_prim(p)/plog(p).

The sum

X

χodd

<Φ∗Φ^∨, χ > χ_prim(p)/p 1−χ_prim(p)/p

is an algebraic number. By construction, it is invariant under the action of any element of Gal(Q|Q) and it is thus an element ofQ. Taking this into account